Prove 1 is not the largest integer? This proof looks extremely flawed, but I'm new to proofs so I'm not completely sure what is allowed and what isn't. Here it is:
Let $n$ be the largest positive integer. Then $n$ must be $\geq 1$. Multiplying both sides of the inequality gives $n^2\geq n$. But, because $n$ is the largest possible integer, we also have $n^2\leq n$, meaning $n^2=n$. Dividing both sides by $n$ gives us $n = 1$.
I'm guessing it has something to do with the $\geq$ sign at the beginning. After all, if $n$ is the largest possible integer, then $n > 1$. But there's also the fact that there is no largest integer. So if I just added "$2 > 1$" to the end of the proof, wouldn't that disprove $1$ being the largest integer?
 A: The prove flawed in its very first line.
Let $ n $ be the largest positive integer.
How do you know a largest positive integer exists at all?
A proper proof involves a definition of integer. Without that there is no hope for a proof.
As Stephen Hawking put it, 'god created the integers', so it is actually kind of hard to have a definition of integer. 
Let's start with something simpler called inductive numbers.
The set of inductive numbers $ I $ is defined as follow:
$ 1 \in I $
If $ n \in I $, then $ n + 1 \in I $.
To give the set an ordering, we define $ n + 1 > n $.
So now it is possible to create a proof that 1 is not the greatest positive integer.
1 + 1 = 2 is by definition in I and is larger than 1, so it is not the greatest positive integer.
I greatly simplified this by picking a very convenient definition. But hopefully this does point out that without a definition of integer you can't really prove anything about your proposition.
For your reference, I found this definition of integer on proof wiki.
https://proofwiki.org/wiki/Definition:Integer
